Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60388 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C210 |
| direct product | C7×#60390 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C22×#120 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C7×#41 ◁ G | |
4698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3,4,5,6,7)(9,14)(12,23)(15,28)(16,27)(17,24)(19,25)(22,34)(26,30)(29,38)(31,35)(33,36)(37,39);(8,9)(10,15)(11,16)(12,24)(13,17)(14,23)(18,29)(19,26)(20,30)(21,31)(22,35)(25,28)(27,34)(32,39)(33,37)(36,38);(8,10,13,20)(9,15,17,30)(11,18,21,32)(12,25,23,19)(14,26,24,28)(16,29,31,39)(22,36,34,33)(27,37,35,38);(8,11)(9,16)(10,18)(12,22)(13,21)(14,27)(15,29)(17,31)(19,33)(20,32)(23,34)(24,35)(25,36)(26,37)(28,38)(30,39);(11,21)(16,31)(18,32)(22,34)(27,35)(29,39)(33,36)(37,38)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53378);